The state of the art in Dynamic Relaxation methods for ...

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Iranian Journal of Numerical Analysis and OptimizationVol. 7, No. 2, (2017), pp 87–114DOI:10.22067/ijnao.v7i2.67470

The state of the art in DynamicRelaxation methods for structural

mechanicsPart 2: Applications

M. Rezaiee-pajand∗, J. Alamatian, and H. Rezaee

Abstract

The most significant advances over the sixty years relative to the DynamicRelaxation applications in structural engineering are summarized. Empha-sized are given towards plates, form finding, cable structures, dynamic analy-

sis, and other applications. The role of DR solver in the linear and non-linearanalyses is discussed. This investigation is undertaken to explain the applica-tion of the solution technique to the static, dynamic and stability problems.The influence of the methods on the use of isotropic and composite materials,

such as orthotropic and laminated ones, is briefly covered. Critical analysesand suggestions regarding future research and applications will be presented.

Keywords: Dynamic Relaxation method; Application; Survey; Solver; Re-view; Iterative technique; State of the art.

1 Introduction

The first part of the paper entitled “The state of art in Dynamic Relaxationmethods” devoted entirely to the DR formulations and its fictitious param-

∗Corresponding author

Received 25 November 2016; revised 12 May 2017; accepted 23 April 2017M. Rezaiee-pajandDepartment of Civil Engineering, Factualy of Engineering, Ferdowsi Unversity of Mashhad,Mashhad, Iran. e-mail: [email protected]

J. AlamatianCivil Engineering Department, Mashhad Branch, Islamic Azad Unversity, Mashhad, Iran.

e-mail: [email protected]

H. Rezaee

Department of Civil Engineering, Factualy of Engineering, Ferdowsi Unversity of Mashhad,Mashhad, Iran.

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88 M. Rezaiee-pajand, J. Alamatian, and H. Rezaee

eters. For this purpose, the main two categories of the DR methods wasreviewed; i.e., the viscous damping and the kinetic damping formulations. Inspite of several similarities in formulation and iterative relationships, the vis-cous and kinetic approaches have the different physical basis. Furthermore,the proposed formulations for fictitious parameters are different. However,the parameters such as fictitious time step and initial displacement haveunique form in both methods.It is worth emphasizing that these solvers have been exploited in a vast areaof the structural analyses. In other words, most of the DR research topicsconducted so far have been related to their applications. There is only a littlefraction of the papers which deal with DR formulations and their efficiencies,such as accuracy, stability and convergence rate. Therefore, the importantissues of the DR applications require special consideration. Based on thisfact, the applications of the DR schemes in the different structural analysisare reviewed. It should be noted that these strategies have been used for thedifferent kind of analyses, such as static, dynamic, stability, form-finding and,etc. Verities of investigators have found the solution of diverse structures,like: truss, frame, shell, plate and, etc. The reasons for these wide rangesof applications can be found in some interesting DR specifications, whichcreate unique potentials in engineering problems. The main specifications ofDR algorithms will be listed in the following lines:

(a) Full vector operations

(b) Calculations are based on internal force rather than the secant stiffnessmatrix

(c) Simple numerical algorithm

(d) Iterative process

(e) Not sensitive to critical points in structural behavior

(f) Full automatic procedure (personal judgments have no effect on thesolution efficiency)

The aforementioned advantages cause numerous usages in different structuralengineering problems. The existing restrictions have not affected the vastapplications of these solvers. So far, the analysts have found that there areverities of interesting and attractive applications of the DR methods. In thefollowing sections, some of them are briefly reviewed.

2 Plates

Rushton applied Dynamic Relaxation to non-linear systems by analyzing vari-able thickness plates with geometrical non-linear behavior [101]. He also used

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The state of the art in Dynamic Relaxation methods ... 89

this technique for solving the large deflection equations resulted from ana-lyzing the post-buckling of variable thickness plates [103]. In another paper,this researcher analyzed the small and large deflection of simply supportedplates with corners free to lift [104]. Moreover, DR method was utilized toanalyze the post-buckling of variable thickness flat rectangular plates withmixed boundary conditions by Rushton [105]. This investigator used theDR method to study the large deflection of plates with small initial curva-ture [106]. By utilizing of the DR technique, Rushton found large deflectionof rectangular plates with unsupported edges [107]. In another research, healso examined the buckling of laterally loaded plates having initial curva-ture [108].

In 1972, Alami analyzed the elastic large deflection behavior of plates un-der transverse loading by a method based on Dynamic Relaxation [1]. TheDR approach was applied to non-linear analysis of clamped orthotropic skewplates having a constant thickness subjected to a uniformly-distributed trans-verse load by Alwar and Ramachandra [5]. Turvey and Wittrick utilized theDR method to solving flexural and stability problems of elastic geometricallybehavior of laminated plates [158]. In another study, this solver was used toanalyze the large deflection of skew sandwich plates with clamped edges byAlwar and Ramachandra [6].

Tuma and Galletly described the use of Dynamic Relaxation for analyzinglarge-deflection of circular and rectangular plates with various boundary con-ditions [128]. This technique was also employed for the analysis of non-linearbehavior of clamped isotropic skew plates of constant thickness subjectedto a uniformly distributed transverse load by Alwar and Rao [7]. Moreover,Murthy et al. adopted the DR scheme for analyzing the non-linear bending ofcircular plates of the linear elastic material and variable profiles subjected toa uniformly distributed lateral loading [78]. Dowling and Bawa used DynamicRelaxation to solve the finite difference form of orthotropic small deflectionplate equations. They took advantage from an analytical method of pro-ducing influence surfaces for stiffened steel bridge decks [33]. Utilizing thisiterative approach, Turvey found the large deflections of square plates, obey-ing the Foeppl and Von Karman theories [129]. In another study, Chaplinused the DR approach to analyze the large elastic deformations of clampedskewed plates [21].

Turvey calculated the large deflection of circular plates with variable thick-ness utilizing two different DR methods. In the first technique, fictitiousdensities were chosen arbitrarily and in the second one, they were calculatedusing Gerschgorin bounds. By comparing the results, it was demonstratedthat the latter improves the convergence characteristics. This researcheralso implemented the second procedure for solving variable thickness annularplates subjected to a uniform lateral load [130]. In the other application,Frieze et al. took advantage of the DR algorithm to analyze the large de-flection of elasto-plastic plates. They extended the strategy to include bothgeometrical and material non-linear effects in plates. Particularly, these in-

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vestigators discussed the iteration parameters which control convergence [37].Frieze and Dowling proposed a process based on Dynamic Relaxation for theexact interactive buckling analysis of plates forming box sections subjectedto a generalized loading [38]. Using the DR approach, Turvey analyzed theaxisymmetric large deflection behavior of circular plates with linear thicknesstapers and affine imperfections [131]. He also applied the mentioned strategyto the analysis of the axisymmetric, elasto-plastic large deflection response oflaterally loaded circular plates [132]. Moreover, Turvey utilized the methodof Dynamic Relaxation for solving the set of non-incremental equations de-scribing the axisymmetric, elasto-plastic, large deflection response of laterallyloaded, thickness-tapered, circular plates [133].

There have been a lot of more studies related to the plate problems. Sher-bourne and Murthy applied the DR method to analyze bending of circularplates with variable profile [120]. Webb and Dowling utilized Dynamic Re-laxation to solve the governing equations from the material and geometricnon-linear analysis of eccentrically stiffened plates, subjected to the combinedor separate actions of lateral and in-plane loading [161]. By using this solver,Turvey analyzed the axisymmetric elastic large deflection behavior of glass-steel and glass-aluminum circular plates [137]. He also employed this kindof algorithm for analyzing axisymmetric small deflection equations for sheardeformation of circular plates [138]. It should be added that Turvey and Limapplied several refinements to the DR method in order to improve computa-tional efficiency. Then, they solved the equations coming from the analysisof axisymmetric full-range response of transverse pressure loaded, clampedand simply supported, circular plates [147]. Turvey and Der Avanessian usedthe DR method for a numerical solution of the governing equations for theaxisymmetric elastic large deflection analysis of ring-stiffened plates [140].In another research, Lim and Turvey adopted the DR approach to solve theplate equations for examining the elastic integrity degradation of uniform flatsimply supported, and clamped circular plates subjected to uniform trans-verse pressure [148]. They also utilized this way in solution of the incrementalequations governing the axisymmetric elasto-plastic large deflection responseof circular steel plates with initial deflections [71].

Numerical solutions of the equations governing the elastic post-buckling re-sponse of flat and imperfect polar orthotropic circular and annular plateswas carried out using a finite-difference version of the DR algorithm by Tur-vey and Drinali [145]. By utilizing weighting factors for mass and damping,Al-Shawi and Mardirosian used Dynamic Relaxation in the finite elementanalysis of plate bending [118]. Method of Dynamic Relaxation was suc-cessfully exploited to large deflection analysis of circular plates by Turveyand Der Avanessian [141]. Turvey and Lim utilized the DR procedure innumerical solution of the incremental equations governing the full-range ax-isymmetric flexural response of thickness-tapered circular plates [149]. Forensuring the numerical stability of DR procedure, Xiao-qing and Wan-linproposed a method for calculation of fictitious densities. Then, they applied

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the DR technique to the non-linear bending problems of rectangular plateslaminated of bimodular composite materials [164]. Machacek employed thissolution process in the analysis of thin steel isotropic, and stiffened platesloaded in compression and bending [73]. In the other study, Turvey andOsman utilized the DR method in solution of first-order orthotropic sheardeformation equations for the non-linear elastic bending response of rectan-gular plates [150].

Using Dynamic Relaxation, Machacek analyzed large deflection elasto-plasticbehavior of thin steel eccentrically stiffened plates subjected to the lateraland/or in-plane load [74]. The large-deflection elasto-plastic behavior of flatplates and attached flat-bar stiffeners, including the effects of interaction wasalso presented using Dynamic Relaxation by Caridis and Frieze [16]. Tur-vey and Der Avanessian utilized the DR method in solution of the govern-ing system of equations for the axisymmetric elasto-plastic large deflectionanalysis of ring-stiffened circular plates [143]. The application of the Dy-namic Relaxation to study the post-buckling path of cylindrically curvedpanels of laminated composite materials during loading and unloading wasalso described by Wan-lin and Xiao-qing [160]. Using the DR techniquein conjunction with graded finite-differences for a numerical solution of thestiffened plate equations, Turvey and Der Avanessian suggested a discretelystiffened plate theory for the elasto-plastic large deflection analysis of pres-sure loaded ring-stiffened circular plates [144]. In the other research, theDR procedure was coupled with an interlacing central finite-difference dis-cretization scheme by Turvey and Salehi. They utilized the aforementionedway in solving equations for elastic large deflection of uniformly loaded sec-tor plates [153]. A finite-difference implementation of the DR process wasdeveloped to solve the non-axisymmetric elastic large deflection equations ofannular sector plate by Salehi and Turvey [114]. In another study, Turveyand Osman utilized finite differences in conjunction with the DR procedurefor numerical solutions of the Mindlin laminated plate equations in large de-flection initial failure analysis of square angle-ply laminated plates subjectedto uniform pressure loading [151].

Combining DR scheme with finite-difference discretization, Salehi et al. an-alyzed geometrically non-linear behavior of sector plates [111]. In the otherstudy, Salehi and Shahidi utilized the mentioned technique for solving thegoverning equations for the elastic large deflection of uniformly pressureloaded sector Mindlin plates [112]. Caridis employed the DR method on aninterlacing finite difference grid for solving the large-deflection elasto-plasticequations of equilibrium of flat and stiffened plates [17]. Turvey and Salehiproposed a discretely stiffened circular plate theory. They obtained the solu-tion of the large deflection plate equations with the aid of a finite-differenceimplementation of the DR algorithm [154]. The large deflection axisymmetriccircular plate equations were solved using a finite-difference implementationof the DR method by Turvey and Drinali [146]. In the other application,Kadkhodayan et al. utilized this solution approach to predict the bifurcation

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point of elastic plates subjected to in-plane and transverse external forces [56].Turvey and Salehi described a theory for the non-axisymmetric elastic largedeflection analysis of sector plates stiffened by a single eccentric rectangularcross-section radial stiffener. They used the DR procedure for solving thediscrete system of equations [155]. Using a finite difference implementationof the DR scheme, Turvey and Salehi also solved the governing equationsfor an elasto-plastic large deflection analysis of pressure loaded sector plates.The material of structure obeyed the Ilyushin full-section yield criterion andthe flow theory of plasticity [156].

Kadkhodayan investigated the implementation of interlacing, and irregularrectangular meshes used in conjunction with the DXDR method, which isa modified DR technique, to study the large deflection behavior of elasticplates [53]. Moreover, this solver together with the finite difference dis-cretization technique were utilized in analyzing the plate structure for ge-ometrically linear and non-linear elastic. The plate material was assumedto be thick composite sector plates by Salehi and Sobhani [113]. Salehi andAghaei took advantage of the DR technique together with the finite differ-ence discretization method in non-linear shear analysis of non-axisymmetriccircular viscoelastic plates [110]. The elasto-plastic large deflection equationsfor annular sector plates were solved by the DR procedure. This study wasperformed by Turvey and Salehi [157]. Furthermore, the DXDR method,which is a modified DR strategy, was utilized for solving equations of largedeflection of rectilinear orthotropic square and circular plates by Kadkho-dayan [54]. In another investigation, combination of the DR method withthe finite difference technique was used to solve the discretized equations inthe analysis of the non-linear bending of annular functionally graded plate(FGM) under mechanical loading [41]. In a recent research, Rezaiee and Es-tiri graded sixteen well-known DR procedures based on their performance interms of the number of iterations and the analysis time in solving severalbending plate examples with geometrical nonlinear behavior [96].

3 Form-finding

One of the extensive uses of DR algorithms is in the structural form-finding.In 1988, Barnes performed the form-finding analysis and fabrication pattern-ing of wide span cable net and pre-stressed membrane roofs by taking advan-tage of the DR method with kinetic damping [9]. A procedure for numericalform-finding of the stable minimal surfaces represented by soap-film models,based on the DR technique, was also suggested by Lewis and Gosling [65].In another study, Gosling and Lewis applied the matrix-based finite elementmethod to the vector-based DR algorithm for form-finding of minimal surfacemembranes [42]. Moreover, the form-finding of lightweight tension structurescomprising pre-stressed cable nets and fabric membranes was performed by

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Lewis WJ and Lewis TS. They utilized DR technique for solving the resultingnon-linear set of equilibrium equations [66]. Adriaenssens and Barnes devel-oped a Dynamic Relaxation based method for form-finding and load analysisof a class of tensegrity structures with continuous tubular compression boomsforming curved splines. This structure could be deployed from straight bypre-stressing a cable bracing system [2].

In the other research, Zhang and Shan utilized the DR method for form-finding of the pre-stressed cable and membrane structures [166]. By DynamicRelaxation with kinetic damping, Brew and Lewis presented a computationalmethodology for form-finding of tension membrane structures (TMS), or fab-ric structures, used as roofing forms [13]. Li and Wei used the DR methodin the formulation of the form-finding under uniform pre-tensioned force andthe behavior analysis under static loads [70]. Han et al. proposed a form-finding strategy for tension structures with elastic supports with the aid ofDR method [43]. The aforementioned algorithm was also applied to form-finding of a grid shell in composite materials by Douthe et al. They modeledthe large structural rotations by means of the DR technique [31]. It shouldbe noted that this kind of rotations may occur by bending of pre-stressedconstructions during the erection process. Furthermore, the form-finding ofnon-regular tensegrity structures by the aid of a scheme, which was basedon the DR method with kinetic damping, and was described by Zhang etal. [167].

Using the DR process, Baudriller et al. suggested a form-finding method forgenerating non-regular tensegrity shapes of higher diversity and complexity[11]. Lu and Ren utilized the DR method for the form-finding of three-dimensional cable structures in simulation of the cable mesh reflector for thelarge radio telescope [72]. In another investigation, Wu and Deng exploitedthe DR method for form-finding of slack cable-bar assembly [162].

Douthe and Baverel proposed a simple way using the DR algorithm to findthe form and perform a structural analysis of multi-reciprocal frame systems.This technique is also called nexorades [30]. Furthermore, Ye and Zhoumodified the DR method in two aspects. The first was a simple procedure togain the damp value directly, which is close to that of the critical damp. In thesecond feature is an extension of the classic DR scheme, which was suggested[165] to obtain the desired membrane shapes. In another research, Lee andHan use DR method for geodesic shape finding of membrane structure withgeodesic string [62].

4 Cable structures

The DR strategy is suitable for finding the response of the flexible structures,such as cable. This solution technique was employed for the analysis of ca-ble structures by Day and Bunce [28]. Moreover, Lewis et al. developed a

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DR method for the analysis of the non-linear static response of pre-tensionedcable roofs [68]. Using the aforementioned procedure, Lewis et al. inves-tigated cladding-network stiffening effects in pre-tensioned cable roofs [69].In another study, Lewis and Shan obtained the solution of the governingequilibrium and compatibility equations in the analysis of cladding-networkinteraction in pre-tensioned cable roof structures under static loads with theaid of the DR scheme with kinetic damping [67]. The DR method was alsoused in the analysis of cable net systems with many loads relieving devicesby Shan et al. [117]. Form-finding, analysis and fabrication patterning ofwide-span cable nets and grid shells, uniform or variably pre-stressed fabricmembranes and battened membrane roofs was carried out using a DynamicRelaxation with kinetic damping based technique by Barnes [10]. Chen etal. applied DR method to static analysis of cable nets [25]. They also em-ployed this solver for finding equilibrium state of construction procedures ofcable-domes by controlling the original length simulation of the constructionprocess [24]. In order to analyze the nonlinear behavior of cable domes, whichis affected by creep over time, a modified dynamic relaxation procedure wasintroduced by Kmet and Mojdis [59].

5 Dynamic analysis

Utilizing DR method in the dynamic structural analysis is a new application,which has been introduced since 2008. There are only a few papers, whichdeal with this interesting application of the DR method, most of them werepresented by Rezaiee and Alamatian [3,93]. In this approach, DR algorithm iscombined with numerical time integrations. It is well known that numericalintegration procedures are appropriate for dynamic analysis of structures.These schemes are commonly applied to both linear and non-linear struc-tural behaviors. The aforementioned techniques can calculate the dynamicresponses via step by step process. Numerical integration strategies are di-vided to the implicit and explicit scheme. In the implicit formulation, thedynamic equilibrium equation of structure at time tk+1is converted to thebelow equivalent static system [3,93]:

[S]k+1EQ {D}

k+1 = {P}k+1EQ (1)

where [S]k+1EQ , {D}k+1 and {P}k+1

EQ are the equivalent stiffness matrix, the

displacement vector and equivalent load vector at k + 1th step of numer-ical integration formulation, respectively. The equivalent stiffness matrixand equivalent load vector are dependent to the implicit dynamic algorithm.Rezaiee-Pajand and Alamatian [93] used the DR method to solve Eq. (1).The proposed technique led to a considerable reduction in errors of non-linear dynamic analysis. Moreover, Sarafrazi presented a different scheme to

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analyze the structures. He assumed real and also fictitious time axes. Inthis strategy, real and fictitious times were considered simultaneously [115].Recently, an explicit dynamic analysis was performed by the Dynamic Re-laxation method [79].

6 Other applications

Because of interesting specifications of the DR method, it has a great poten-tial ability to be used in other kinds of structural engineering problems. Forinstance, Hussain Khan and Lafitte employed a finite difference based strat-egy exploiting the DR procedure for performing design analysis of a cylindri-cal pre-stressed concrete pressure vessel [52]. Moreover, Rushton discussedthe determination of the critical damping and optimum time increment. Heutilized the DR method for stress analysis problems [102]. The DR methodwas also developed for time dependent stress analysis of the pre-stressed con-crete pressure vessels by Cederberg and David [20]. Cassell introduced theconcept of fictitious densities to the DR method. He analyzed doubly curvedshells under arbitrary surface tractions and temperature loading [18]. Fur-thermore, Brew and Brotton formulated the DR method using the first-orderdynamic equilibrium relationship and studied the stability conditions of theframe structures [12]. Shmuel and Alterman proposed a scheme for analyz-ing the dynamic stress distribution close to a crack which cuts through thefinite thickness of a given plate. They utilized the aforementioned techniqueto rapidly improve the convergence of their suggested strategy [121]. TheDR procedure was also applied to the analysis of bridge slabs by Cassell etal. [19].

In another application, Hansson used this solver for calculation of plane prob-lems with general shape and non-linear material behavior [45]. Bunce andBrown utilized the DR approach in solving the equilibrium equations resultedfrom analyzing the finite deflection of plane frames [14]. The DR method wasalso exploited to computation of stresses, and displacements induced by un-derground excavations by Crouch and Fairhurst [27]. In addition, Hanssonand Schnellenbach analyzed some special problems of local stress conditionsof pre-stressed concrete reactor pressure vessels with non-linear material be-havior. They performed calculations by the DR process [46]. By use of thissolution strategy, Rushton and Hook analyzed the large deflections of rectan-gular plates and beams obeying a non-linear stress-strain law [109]. The DRapproach was also used for analyzing disks with anisotropic material, com-plying with Ramberg-Osgood stress-strain relations with variable profiles bySherbourne and Murthy [119]. In the other research, Hook and Rushton in-vestigated beams and plates problems, which deflect until contact is madewith internal restraints, using a modified form of the DR method [51]. Swantook advantage of the DR scheme for the dynamic stress analysis of cer-

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tain linear elastic fracture problems [124]. Moreover, the DR methodologywas utilized in calculations for the analysis of two-dimensional behavior ofthe liners in pre-stressed concrete reactor pressure vessels by Oberpichler etal. [83].

Hansson and Stoever described a process based on the DR algorithm forthree-dimensional analysis of local crack development in pre-stressed concretepressure vessels [47]. On the other hand, Bunce and Brown proposed atechnique based on Dynamic Relaxation for analyzing the non-linear behaviorof large deflection of thin ideally elastic rods [15]. Using the method ofDynamic Relaxation, Frieze and Dowling analyzed the elastic post-bucklinginteraction behavior of box sections [38]. In another investigation, Friezeapplied Dynamic Relaxation to the analysis of interactive buckling in theelasto-plastic range of thin-walled rectangular sections [37]. Harding usedthe DR method to solve governed equations from elasto-plastic analysis ofinitially imperfect thin cylinders subjected to axial compression and externalpressure [48]. The DR method was also utilized to solve the axisymmetric,large deflection response of a uniformly loaded, circular membrane subjectedto an initial pre-stress/pre-strain by Turvey [134]. In another study, Tsotsosand Hatzigogos adopted the DR procedure for the analysis of the foundationsof dams and embankments [127].

Oberpichler performed calculations for designing the liner-anchor systemof concrete vessels by the DR method [81]. Moreover, Turvey combinedthe Dynamic Relaxation with the Tsai-Hill failure criterion and obtaineda non-linear elastic solution for the large deflection of laminated strip equa-tions [135]. In another research, Oberpichler used the DR approach for designcalculations of steel-liners (in Pre-stressed Concrete Reactor Pressure Vessel(PCRV) or a Concrete Containment Vessel (PCCV)) and their anchoragewith regard to non-linear behavior of liner-material and anchorage [82]. Tur-vey also utilized the DR method to solve the equations for anti-symmetricallaminated, cross-ply strips [136]. In the other paper, Papadrakakis developedDR scheme for the post-buckling behavior of spatial structures [84]. Numeri-cal studies were carried out on the elastic large deflection behavior of cross-plylaminated bi-modular strips employing the DR method by Turvey [139]. In1982, an implicit DR relationship exploring general increment control wasintroduced by Felippa [36]. Moreover, Estefen and Harding described a fi-nite difference DR program for the analysis of ring-stiffened shells, includingthe behavior of the rings as discrete elements [34]. The DR method wasalso applied to the solution of the non-linear equilibrium equations governingthe analysis of the post-critical ultimate load conditions of space trusses byPapadrakakis and Manolis [85].

Krings et al. performed the requisite non-linear calculations for the de-sign of concrete tunnel linings with non-linear material behavior by the DRscheme [61]. Zienkiewicz et al. suggested an accelerated procedure for im-provement of the convergence rate [172]. Using the DR technique, a methodwas proposed for finding the optimum layout of modularly constrained truss

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structures subjected to multiple load conditions by Topping [125]. Estefenand Harding also investigated the inter-ring buckling of cylindrical leg mem-bers by use of the DR scheme [35]. Analysis of a small-deflection beam wascarried out with the aid of DR technique by Turvey and Der Avanessian [142].In order to facilitate static analysis of non-linear problems, Rericha modifiedthe DR method. He exploited continuous loading in time instead of the or-dinary step function of time. Inertia and damping forces arising during theloading process were kept at a minimum using an optimum load time his-tory [92]. In another study, Chou and Wu proposed a Dynamic Relaxationfinite element method for metal forming processes [26]. Using a naturalcombination of Lyaponov’s dynamic criterion on stability and the modifiedadaptive DR method, Zhang et al. obtained an approach for predicting thebifurcation points of elastic-plastic buckling of plates and shells [169].

Qiang determined fictitious time and damping by Rayleigh’s principle [87].In another investigation, Lewis used the DR process in the analysis of thenon-linear static response of pre-tensioned cable net and pin-jointed framestructures [64]. The modified adaptive DR method (maDR method) wasapplied to study of the axisymmetric deformation mechanism of work-piecesin conical cup tests by Zhang et al. [170]. Chen et al. also developed anumerical method for modeling plane strain rolling based on DR technique[22]. Furthermore, Rao et al. investigated the use of the DR scheme inthe problems with geometric and material non-linearities. It was found fromnumerical studies that in the case of geometric non-linearity, obtained steady-state solution compares well with the static solution. In this technique, theprocess of convergence to steady state can be accelerated by choosing highermass densities and an appropriate mass proportional damping. On the otherhand, when both non-linearities were existed, diverse steady-state solutionswere obtained for different values of artificial damping [91]. The applicationof the DR method to study the post-buckling path of cylindrically curvedpanels of laminated composite materials during loading and unloading wasalso described by Wan-lin and Xiao-qing [160]. In another paper, Moncrieffand Topping introduced a new procedure for generating cutting patterns formembrane structures, by means of a two-dimensional Dynamic Relaxationanalysis for each cloth [77].

Using an improved adaptive DR method, Zhang analyzed the shape effects onthe tensile strength of axisymmetric rock specimens [168]. Shugar proposedthe automated Dynamic Relaxation (ADR) algorithm. He determined theinitial equilibrium configuration and pre-stressed state of compliant struc-tures by this scheme [123]. Moreover, the application of DR method tomodel discrete crack propagation was described by Gerstle et al. [40]. Sauveand Badie combined the shell formulation with the DR technique, a three-dimensional contact algorithm and generalizations of the constitutive equa-tions for time-dependent deformation to yield accurate simulations of creepresponse in nuclear reactor fuel channels [116]. In an interesting applica-tion, a variable-arc-length approach was introduced by Ramesh and Krish-

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namoorthy, which can trace snap-through or snap-back regions of the load-deflection path, automatically. They proposed a displacement incrementationprocedure to handle multiple loadings in the post-buckling analysis of struc-tures by Dynamic Relaxation [88]. DR approach was also used to analyzeequilibrium configurations of partly wrinkled membranes by Haseganu andSteigmann [49].

New schemes for Elastic Post-Buckling (EPB) analysis and Inelastic Post-Buckling (IEPB) analysis of truss structures using the DR method were intro-duced by Ramesh and Krishnamoorthy. These researchers used the variable-arc-length technique for tracing the post-buckling paths for elastic, EPB andIEPB analyses [89]. In the other research, a parallel algorithm for the DRstrategy was presented by Topping and Khan [126]. Moreover, Kommineniand Kant described a unified approach for linear and geometrically non-linearanalyses of composite and sandwich shells by means of the DR process [60].The DR iteration was also adopted for the geometrically non-linear analysisof plates and shells involving large deflections, small rotations and strains byRamesh and Krishnamoorthy [90]. Kadkhodayan and Zhang combined twovital components: the DXDR algorithm and a stable consistent algorithmfor the integration of the constitutive equations of plasticity. They used thisstrategy for analyzing general elastic-plastic problems [55]. Furthermore, theDR method was adopted for solving delayed hydride cracking (DHC) prob-lems by Metzger and Sauve [75].

Using the DR process, Turvey and Osman analyzed small and large deflectionMindlin plate equations for unsymmetrical laminated cross-ply strips sub-jected to uniform lateral pressure [152]. Oakley and Knight Jr suggested aparallel adaptive Dynamic Relaxation (ADR) algorithm for non-linear struc-tural analysis [80]. By means of finite differences together with DR method,Atai and Steigmann developed an equilibrium theory for the coupled finitedeformations of elastic curves and surfaces [8]. The DR procedure was alsoemployed in the analysis the fracture of materials by Metzger [76]. More-over, Wakefield presented the application of DR approach to the analysis oftensile structures [159]. Xiaoqing and Hong utilized the DR solver to inves-tigate the influence of compression-bending coupling on non-linear behaviorof cylindrically slightly curved panels of unsymmetrical laminated compos-ite materials subjected to uniform uniaxial compression during loading andunloading [163]. Han and Lee employed DR procedure for stabilizing of un-stable structures [44]. In addition, the Dynamic Relaxation was combinedwith neural networks to increase model accuracy of tensegrity structures byDomer et al. [29]. Shoukry et al. employed DR approach for analyzing largetransportation structures as dowel jointed concrete pavements and 306-m-long, reinforced concrete bridge superstructure under the effect of temper-ature variations [122]. In another study, Chen et al. used the DR methodand obtained a technique to find the internal force of the elements, and thenodal coordinates of the structures composed of cables in tension and strutsin compression in the equilibrium state by controlling the original length [23].

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Hegyi et al. employed the DR method for membrane analysis by an eight-node quadrilateral element [50]. Kan and Ye described a way based on mod-ified Dynamic Relaxation for solving the problem of rigid displacement [57].In another study, Douthe et al. exploited the DR method in numerical mod-eling of grid shell with composite materials [32]. Plotzitza et al. employedthe DR technique in the numerical simulation of concrete under explosiveloading [86]. Moreover, the DR method was utilized to obtain the analyticalsolution of free damped vibrations of a linear viscoelastic oscillator, based onthe fractional derivative model involving more than one different fractionalparameter and several relaxation (retardation) times by Rossikhin et al. [100].For reducing the calculation cost of the DR technique, Kashiwa and Onodacombined this method with an iterative technique via two switching rules.In this strategy, the numerical scheme used in the analysis is selected as thesituation of the analysis. An unstable region of the analysis, in which thestatic iterative method cannot obtain the converged solution, is performedby the robust DR method. Then, a stable region that is costly to be solvedby the DR technique is performed by the efficient iterative method. Theperformance of the suggested process was verified through a comparison ofnumerical analyses for the wrinkled membrane with the new way and con-ventional ones [58].

Based on the fact that a static problem has an equivalent wave speed of in-finity, and a dynamic problem has a wave speed of finite value, an effectiveloading algorithm associated with the explicit DR procedure was presentedto produce meaningful numerical solutions for static problems by Zhao etal. [171]. Rezaiee-Pajand and Alamatian proposed a method for tracingstatic path of truss structures using DR technique [94]. Furthermore, anautomatic DR method was proposed for tracing the static path of structures,which have snap-through or snap-back regions, by these researchers [95]. Ina recent research by Rezaiee and Estiri, a variable load factor, obtained fromminimization of the unbalanced displacement increment, was employed in DRalgorithm. Consequently, their proposed process was capable of tracing theequilibrium path at snap-through and snap-back points in addition to pos-sessing high accuracy and efficiency [96]. In another paper, they minimizedthe work increment by the external loads and obtained a new formulation ofthe load factor [96]. Utilizing the structural equilibrium path, which was ob-tained from the DR method, these researchers calculated the buckling limitload. They set the work increment of the external forces to zero and founda new equation to derive the load factor [96]. Moreover, the first structuralbuckling load was recently calculated by simple DR algorithm [4]. Lee etal. presented an explicit arc-length method based on cylindrical arc-lengthmethod using Dynamic Relaxation with kinetic damping for tracing the post-buckling equilibrium path of structures [63].

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GalleyProof


7 Conclusion

This paper was devoted to the applications of the DR methods. Accord-ing to all the evidence presented throughout this study, these techniques fitquite well with the linear and also non-linear behaviors of structures. Theywere successfully utilized in static and dynamic problems. A variety of thestructures, such as trusses, frames, plates, membranes, shells and cables, canbe analyzed by the aforementioned formulations. This survey showed thatthe applications of DR methods in structural engineering problems are veryvast. The enormous usage of these solvers in different branches of structuralanalysis is due to simplicity, full explicit vector operations and no sensitivityto non-linear behaviors, which are the main specifications of DR procedure.

In this paper, a wide range of DR applications was analyzed, and very usefulreference sources were introduced. Due to important roles of these strategies,providing a reasoned and evidence critique and also more practical waysof function are still remained to be found. Based on the broad featuresof presented studies, it is suitable to make some suggestions for the futureresearch and applications of these non-linear solvers. From the relevancepoint of view, the DR method has a great potential of abilities to be use indifferent engineering problems. These applications could not be limited tothe structural analysis, so that the DR algorithms may be utilized for otherbranches of civil engineering, such as hydraulics, numerical solution of waterflows, and soil mechanics. It should be noted that there in no limitation forusing DR tactic in the other engineering and applied mathematics branches.

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مکانیکی های سازه در پویا رهایی های روش بر مروریکاربردها : دوم بخش

رضایی۳ حسینه و علامتیان۲ جواد پژند۱، رضایی محمد

عمران مهندسی گروه مهندسی، دانشکده مشهد، فردوسی دانشگاه ۱

عمران مهندسی گروه مشهد، واحد اسلامی، آزاد دانشگاه ۲

عمران مهندسی گروه مهندسی، دانشکده مشهد، فردوسی دانشگاه ۲

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